Colour wheels

We had a good number of solutions sent in. Some sent in just the first part and that's fine - thank you. Others went through to the end.

Elijah who is home educated sent in the following.

RGBRGB wheel

A multiple of $3$ is blue. (You can quickly see if a number is multiple of $3$ by adding its digits and seeing if that sum is a multiple of $3$.)

$1$ more than a multiple of $3$ is red.

$1$ less than (or $2$ more than) a multiple of $3$ is green.

So:

$18$th is blue

$19$th is red

$31$st is red

$59$th is green

$299$th is green

$3311$th is green

$96312$th is blue

BBYG wheel

A multiple of $4$ is green.

So $24$th is green.

$49$ is $1$ more than $48$, which is a multiple of $4$, so $49$th is blue.

$100$th is green.

BYBR wheel

A multiple of $4$ is red.

So $24$th is red.

$49$ is $1$ more than $48$, which is a multiple of $4$, so $49$th is blue.

$100$th is red.

RRRBBY wheel

A multiple of $6$ is yellow.

So $24$th is yellow.

$49$ is $1$ more than $48$, which is a multiple of $6$, so $49$th is red.

$100$ is one more than $99$.

$99$ is an odd multiple of $3$.

Odd multiples of $3$ are red.

Blue comes after red, so $100$th is blue.

RRBRR wheel

There are blue marks at these places:

$3, 8, 13, 18, 23, 28, 33, 38$ ....

If the number ends with $3$ or $8$ there will be a blue mark.

Everything else is red.

So $24$th, $49$th and $100$th are red.

A wheel with six markings

The $6$th colour marking will be on the multiples of $6$.

$102$ is a multiple of $6$.

$100$ is $2$ less than $102$, so red must be $2$ back from the $6$th marking.

So if you want the $100$th mark to be red, the red must be the $4$th colour marking on the wheel.

Other wheels to make a red $100$th mark

How many colour markings: Position of red colour marking:

$5$ ---------------------------------$5$th

$7$ ---------------------------------$2$nd

$8$ ---------------------------------$4$th

$9$ ---------------------------------$1$st

$10$ -------------------------------$10$th

Emma from Bradon Forest School in the U.K. Sent in a similar solution but these were her explanations for the RRBRR wheel.

RRBRR:

Because red appears on every $5$th colour, the pattern is $5, 10, 15, 20$.

The last red of every spin is in the $5$ times table.

As for the other colours you find out how many $5$s there are in the number and then add however many you need to get the number you want. If you need to add $1, 2$ or $4$ then the mark will be red but if you need to add $3$ then the mark will be blue.

E.g to find the $20$th colour you find out how many $5$s are in $20$ which is $4$. $4$ times $5$ = $20$. You don't need to add anything so the mark is red.

The $100$th mark: You divide $100$ by $6$ which = $16$r$4$. $16$ means the wheel would have to spin $16$ times to get to $100$. r$4$ means that red should be the $4$th spoke on it so it can be the $100$th mark.

Well done to the many of you for the ideas you shared and the solutions you sent in - keep them coming.